Given: In trapezium PQRS, sides `PQ||` sides `SR, AR=5AP,AS=5AQ,` then prove that `SR=5PQ`.

Explanation

Side `PQ||` side SR and line QS is the transversal
`:./_PQS~=/_RSQ` ……(Alternate angles)
i.e. `/_PQA~=/_RSA..(Q-A-S)`…….1
In `DeltaPQA` and `DeltaRSA`
`/_PQA~=/_RSA`……[From 1]
`/_PAQ~=/_RAS`. (Vertically opposite angles)
`:.DeltaPQA~DeltaRSA`...(AA test of similarity)
`:.(PQ)/(RS)=(AQ)/(AS)=(AP)/(AR)`
............(Corresponding sides of similar triangles are in proportion) .......2
`AR=5AP`....(Given ) ...........3
Substituting 3 in 2 we get
`:.(PQ)/(SR)=(AQ)/(AS)=(AP)/(5AP)`
`:.(PQ)/(SR)=(AQ)/(AS)=1/5`............4
`:.(PQ)/(SR)=1/5`........[From 4]
`:.SR=5PQ`
Alternative method
`AR=5AP` .(Given)
`:.(AR)/(AP)=5/1` ......1
`AS=5AQ`........(Given)
`:.(AS)/(AQ)=5/1`......2
In `DeltaRAS` and `DeltaPAQ`.
`(AR)/(AP)=(AS)/(AQ)` ......[From 1 and 2]
`/_RAS~=/_PAQ`...[vertically opposite angles]
`:.DeltaRAS~DeltaPAQ`..(SAS test of similarity)
`:.(AR)/(AP)=(SR)/(QP)` ........[Corresponding sides of similar triangles are in proportion]
`:.5/1=(SR)/(QP)` [From 1 ]
`:.SR=5PQ`